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A054980
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Primitive e-perfect numbers: primitive elements of the e-perfect numbers (A054979).
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16
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OFFSET
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1,1
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COMMENTS
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The nonprimitive e-perfect numbers are obtained from the primitive ones by multiplying by m, if m is squarefree and relatively prime to the primitive e-perfect number.
The following numbers also belong to this sequence; however, their actual positions are unknown: 168136940595306022660197936246988800, 11712310558743727210993873194516480000, 1307484087615221689700651798824550400000. - Andrew Lelechenko, Apr 01 2014
The number of terms with a given number of distinct prime divisors is finite (Straus and Subbarao, 1974). - Amiram Eldar, Mar 04 2021
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B17, pp. 110-111.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 116-117.
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LINKS
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EXAMPLE
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180 = 36*5 (nonprimitive). 252 = 36*7 (nonprimitive). 1260 = 36*5*7 (nonprimitive). 1800 = 36*5^2 (primitive, 5^2 not squarefree and coprime to 36).
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PROG
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(PARI) eperfect(n)=my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d))==2*n
is(n)=if(!eperfect(n), 0, my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]==1&&eperfect(n/f[i, 1]), return(0))); 1) \\ Charles R Greathouse IV, Nov 22 2011
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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